Problem: $\dfrac{dy}{dx}=6x$ and $y(2)=8$. $y(4)=$
Solution: The differential equation is separable. What does it look like after we separate the variables? $dy=6x\,dx$ Let's integrate both sides of the equation. $\int dy=\int6x\,dx$ What do we get? $y=3x^2 +C$ What value of $C$ makes the solution curve pass through the point $(2,8)$ ? Let's substitute $x=2$ and $y=8$ into the equation and solve for $C$. $\begin{aligned} 8&=3\cdot2^2 +C\\ \\ 8&=12+C\\ \\ C&=-4 \end{aligned}$ Now use this value of $C$ to find $y$ when $x=4$. $\begin{aligned} y&=3\cdot4^2-4\\ \\ &=48-4\\ \\ &=44 \end{aligned}$